It's quite possible, and some animals already have them.
Some arachnids have what are known as book lungs - small respiratory cavities that actually are not lungs, but different respiratory organs. They formed separately (in terms of evolution), but have taken the place of lungs in these arachnids.
Some organisms with book lungs have them in two pairs, while others have anywhere from one pair to four pairs. The reason for this many is simple: The book lungs are small. That said, arachnids are (in general) quite small, but some of them are still too big to be sustained by only one pair (or one book lung).
What exactly can extra lungs do in creatures that already get enough oxygen with just one pair? Well, you will have to solve the issue of body structure. Extra lungs take up a substantial amount of space, and animals won't necessarily evolve to be bigger, because they might not benefit much from the extra pair.
Let's assume, though, that a creature does have four lungs. What are the effects? You essentially have a creature with a larger lung volume, assuming that the lungs retain their same size - in other words, assuming that you're adding a pair of lungs to a creature which already has one pair. This means it can
- Survive at higher elevations.
- Grow larger/taller.
- Have different voices (see Iwarsson (2001)).
- Be more resistant to effects from diseases like emphysema, that attack the lungs.
Also, lungs do lots of other things besides helping creatures breathe, including
- Regulate blood pressure.
- Balance blood pH.
- Attack blood clots.
- Protect the heart (or hearts, if you're being creative!).
- Act as backup storage for blood in cases of severe bleeding.
All of these capabilities (and more) would be increased with an extra pair of lungs.